Learn English – “How dare you” vs “How to dare”

You could add some additional words to convey the emphasis, or rephrase slightly:

Number 1:

He certainly can run fast.

He can definitely run fast.

Number 2:

Please do come in

Come in, please (more of a command, to provide the additional emphasis).

You may come in now (the emphasis on "may" in the example suggests that the listener wasn't allowed to come in before, but can now).

Number 3:

I really must brush my teeth.

I definitely have to brush my teeth.

Your friend is correct. "I need compute ..." is ungrammatical, but "I need only compute ..." is fine, if a little bit old-fashioned and formal.

Modal verbs do not use a "to". That is, you say

I can do this.

The verb "need" is a funny case; it is only modal in the negative. In the positive, we already have an equivalent modal verb; namely, "I must". However, there are two possible meanings for the opposite of "I must do this": "I am forbidden to do this" and "I am not required to do this". These two different meanings are conveyed by the modal verbs "I must not" and "I need not".

Searching with Google books, it appears to have been this way at least since 1600 (although back then, there was a positive construction "I must needs", which has since for the most part fallen out of use). Thus, you get various grammatical constructions.

In the negative, you have:

I need not do this.
I do not need to do this.

In the positive, you have:

I must do this.
I need to do this.
*I must needs do this. (obsolete)

"I need do this" is incorrect.

Over the last few centuries, "I don't need to" has slowly been replacing "I need not", but "I need not do this" is still used reasonably frequently, and is grammatical. However, if "I need do this" was ever grammatical, it was in the long distant past.

Finally, in the past you could say

it is not the case that I need compute this,

since that is a negative use (and this is why your friend might not be wrong). Today, I believe most people would use "need to" here. But if your friend was using "need compute" in the negative, there is a good case to make for it being grammatical.